# Twist maps as energy minimisers in homotopy classes: symmetrisation and the coarea formula

## TL;DR

This paper studies twist maps as energy minimisers within specific homotopy classes in an annular domain, using symmetrisation and the coarea formula to establish minimality and local minimality results, especially in two dimensions.

## Contribution

It demonstrates the minimality of twist maps as extremisers of a specific energy functional in their homotopy classes, employing symmetrisation and coarea techniques, with extensions to higher dimensions.

## Key findings

- Twist maps are extremisers of the energy functional in their homotopy classes.
- Twist maps are proven to be local minimisers of the energy in 2D.
- Extensions to higher dimensions and related functionals are discussed.

## Abstract

Let $\X = \X[a, b] = \{x: a<|x|<b\}\subset \R^n$ with $0<a<b<\infty$ fixed be an open annulus and consider the energy functional, \begin{equation*} {\mathbb F} [u; \X] = \frac{1}{2} \int_\X \frac{|\nabla u|^2}{|u|^2} \, dx, \end{equation*} over the space of admissible incompressible Sobolev maps \begin{equation*} {\mathcal A}_\phi(\X) = \bigg\{ u \in W^{1,2}(\X, \R^n) : \det \nabla u = 1 \text{ {\it a.e.} in $\X$ and $u|_{\partial \X} = \phi$} \bigg\}, \end{equation*} where $\phi$ is the identity map of $\overline \X$. Motivated by the earlier works \cite{TA2, TA3} in this paper we examine the {\it twist} maps as extremisers of ${\mathbb F}$ over ${\mathcal A}_\phi(\X)$ and investigate their minimality properties by invoking the coarea formula and a symmetrisation argument. In the case $n=2$ where ${\mathcal A}_\phi(\X)$ is a union of infinitely many disjoint homotopy classes we establish the minimality of these extremising twists in their respective homotopy classes a result that then leads to the latter twists being $L^1$-local minimisers of ${\mathbb F}$ in ${\mathcal A}_\phi(\X)$. We discuss variants and extensions to higher dimensions as well as to related energy functionals.